elmrsubrn - Mathbox for Mario Carneiro

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Theorem elmrsubrn 31417

Description: Characterization of the substitutions as functions from expressions to expressions that distribute under concatenation and map constants to themselves. (The constant part uses

$\$

because we don't know that

and

are disjoint until we get to ismfs 31446.) (Contributed by Mario Carneiro, 18-Jul-2016.)

**Hypotheses**
Ref	Expression
mrsubccat.s	mRSubst
mrsubccat.r	mREx
mrsubcn.v	mVR
mrsubcn.c	mCN

**Assertion**
Ref	Expression
elmrsubrn	$/\$ $\$ $/\$ ++ ++

Distinct variable groups:

Allowed substitution hints:

(

)

(

)

(

)

Proof of Theorem elmrsubrn Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mrsubccat.s	. . . 4 mRSubst
2		mrsubccat.r	. . . 4 mREx
3	1, 2	mrsubf 31414	. . 3
4		mrsubcn.v	. . . . 5 mVR
5		mrsubcn.c	. . . . 5 mCN
6	1, 2, 4, 5	mrsubcn 31416	. . . 4 $/\$ $\$
7	6	ralrimiva 2966	. . 3 $\$
8	1, 2	mrsubccat 31415	. . . . 5 $/\$ $/\$ ++ ++
9	8	3expb 1266	. . . 4 $/\$ $/\$ ++ ++
10	9	ralrimivva 2971	. . 3 ++ ++
11	3, 7, 10	3jca 1242	. 2 $/\$ $\$ $/\$ ++ ++
12	5, 4, 2	mrexval 31398	. . . . . . 7 Word
13	12	adantr 481	. . . . . 6 $/\$ $/\$ $\$ $/\$ ++ ++ Word
14		s1eq 13380	. . . . . . . . . . . . 13
15	14	fveq2d 6195	. . . . . . . . . . . 12
16		eqid 2622	. . . . . . . . . . . 12
17		fvex 6201	. . . . . . . . . . . 12
18	15, 16, 17	fvmpt 6282	. . . . . . . . . . 11
19	18	adantl 482	. . . . . . . . . 10 $/\$ $/\$ $\$ $/\$ ++ ++ $/\$ $/\$
20		difun2 4048	. . . . . . . . . . . . . . 15 $\$ $\$
21	20	eleq2i 2693	. . . . . . . . . . . . . 14 $\$ $\$
22		eldif 3584	. . . . . . . . . . . . . 14 $\$ $/\$
23	21, 22	bitr3i 266	. . . . . . . . . . . . 13 $\$ $/\$
24		simpr2 1068	. . . . . . . . . . . . . 14 $/\$ $/\$ $\$ $/\$ ++ ++ $\$
25		s1eq 13380	. . . . . . . . . . . . . . . . 17
26	25	fveq2d 6195	. . . . . . . . . . . . . . . 16
27	26, 25	eqeq12d 2637	. . . . . . . . . . . . . . 15
28	27	rspccva 3308	. . . . . . . . . . . . . 14 $\$ $/\$ $\$
29	24, 28	sylan 488	. . . . . . . . . . . . 13 $/\$ $/\$ $\$ $/\$ ++ ++ $/\$ $\$
30	23, 29	sylan2br 493	. . . . . . . . . . . 12 $/\$ $/\$ $\$ $/\$ ++ ++ $/\$ $/\$
31	30	anassrs 680	. . . . . . . . . . 11 $/\$ $/\$ $\$ $/\$ ++ ++ $/\$ $/\$
32	31	eqcomd 2628	. . . . . . . . . 10 $/\$ $/\$ $\$ $/\$ ++ ++ $/\$ $/\$
33	19, 32	ifeqda 4121	. . . . . . . . 9 $/\$ $/\$ $\$ $/\$ ++ ++ $/\$
34	33	mpteq2dva 4744	. . . . . . . 8 $/\$ $/\$ $\$ $/\$ ++ ++
35	34	coeq1d 5283	. . . . . . 7 $/\$ $/\$ $\$ $/\$ ++ ++
36	35	oveq2d 6666	. . . . . 6 $/\$ $/\$ $\$ $/\$ ++ ++ freeMnd _g freeMnd _g
37	13, 36	mpteq12dv 4733	. . . . 5 $/\$ $/\$ $\$ $/\$ ++ ++ freeMnd _g Word freeMnd _g
38		elun2 3781	. . . . . . . 8
39		simpr1 1067	. . . . . . . . . 10 $/\$ $/\$ $\$ $/\$ ++ ++
40	39	adantr 481	. . . . . . . . 9 $/\$ $/\$ $\$ $/\$ ++ ++ $/\$
41		simpr 477	. . . . . . . . . . 11 $/\$ $/\$ $\$ $/\$ ++ ++ $/\$
42	41	s1cld 13383	. . . . . . . . . 10 $/\$ $/\$ $\$ $/\$ ++ ++ $/\$ Word
43	12	ad2antrr 762	. . . . . . . . . 10 $/\$ $/\$ $\$ $/\$ ++ ++ $/\$ Word
44	42, 43	eleqtrrd 2704	. . . . . . . . 9 $/\$ $/\$ $\$ $/\$ ++ ++ $/\$
45	40, 44	ffvelrnd 6360	. . . . . . . 8 $/\$ $/\$ $\$ $/\$ ++ ++ $/\$
46	38, 45	sylan2 491	. . . . . . 7 $/\$ $/\$ $\$ $/\$ ++ ++ $/\$
47	15	cbvmptv 4750	. . . . . . 7
48	46, 47	fmptd 6385	. . . . . 6 $/\$ $/\$ $\$ $/\$ ++ ++
49		ssid 3624	. . . . . 6
50		eqid 2622	. . . . . . 7 freeMnd freeMnd
51	5, 4, 2, 1, 50	mrsubfval 31405	. . . . . 6 $/\$ freeMnd _g
52	48, 49, 51	sylancl 694	. . . . 5 $/\$ $/\$ $\$ $/\$ ++ ++ freeMnd _g
53		fvex 6201	. . . . . . . . . 10 mCN
54	5, 53	eqeltri 2697	. . . . . . . . 9
55		fvex 6201	. . . . . . . . . 10 mVR
56	4, 55	eqeltri 2697	. . . . . . . . 9
57	54, 56	unex 6956	. . . . . . . 8
58	50	frmdmnd 17396	. . . . . . . 8 freeMnd
59	57, 58	ax-mp 5	. . . . . . 7 freeMnd
60	59	a1i 11	. . . . . 6 $/\$ $/\$ $\$ $/\$ ++ ++ freeMnd
61	57	a1i 11	. . . . . 6 $/\$ $/\$ $\$ $/\$ ++ ++
62	45, 43	eleqtrd 2703	. . . . . . 7 $/\$ $/\$ $\$ $/\$ ++ ++ $/\$ Word
63		eqid 2622	. . . . . . 7
64	62, 63	fmptd 6385	. . . . . 6 $/\$ $/\$ $\$ $/\$ ++ ++ Word
65	13, 13	feq23d 6040	. . . . . . . 8 $/\$ $/\$ $\$ $/\$ ++ ++ Word Word
66	39, 65	mpbid 222	. . . . . . 7 $/\$ $/\$ $\$ $/\$ ++ ++ Word Word
67		simpr3 1069	. . . . . . . 8 $/\$ $/\$ $\$ $/\$ ++ ++ ++ ++
68		simprl 794	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
69	12	adantr 481	. . . . . . . . . . . . . . . 16 $/\$ Word
70	69	adantr 481	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ Word
71	68, 70	eleqtrd 2703	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ Word
72		simprr 796	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
73	72, 70	eleqtrd 2703	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ Word
74		eqid 2622	. . . . . . . . . . . . . . . . . 18 freeMnd freeMnd
75	50, 74	frmdbas 17389	. . . . . . . . . . . . . . . . 17 freeMnd Word
76	57, 75	ax-mp 5	. . . . . . . . . . . . . . . 16 freeMnd Word
77	76	eqcomi 2631	. . . . . . . . . . . . . . 15 Word freeMnd
78		eqid 2622	. . . . . . . . . . . . . . 15 freeMnd freeMnd
79	50, 77, 78	frmdadd 17392	. . . . . . . . . . . . . 14 Word $/\$ Word freeMnd ++
80	71, 73, 79	syl2anc 693	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ freeMnd ++
81	80	fveq2d 6195	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ freeMnd ++
82		ffvelrn 6357	. . . . . . . . . . . . . . 15 $/\$
83	82	ad2ant2lr 784	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
84	83, 70	eleqtrd 2703	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ Word
85		ffvelrn 6357	. . . . . . . . . . . . . . 15 $/\$
86	85	ad2ant2l 782	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
87	86, 70	eleqtrd 2703	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ Word
88	50, 77, 78	frmdadd 17392	. . . . . . . . . . . . 13 Word $/\$ Word freeMnd ++
89	84, 87, 88	syl2anc 693	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ freeMnd ++
90	81, 89	eqeq12d 2637	. . . . . . . . . . 11 $/\$ $/\$ $/\$ freeMnd freeMnd ++ ++
91	90	2ralbidva 2988	. . . . . . . . . 10 $/\$ freeMnd freeMnd ++ ++
92	69	raleqdv 3144	. . . . . . . . . . 11 $/\$ freeMnd freeMnd Word freeMnd freeMnd
93	69, 92	raleqbidv 3152	. . . . . . . . . 10 $/\$ freeMnd freeMnd Word Word freeMnd freeMnd
94	91, 93	bitr3d 270	. . . . . . . . 9 $/\$ ++ ++ Word Word freeMnd freeMnd
95	94	3ad2antr1 1226	. . . . . . . 8 $/\$ $/\$ $\$ $/\$ ++ ++ ++ ++ Word Word freeMnd freeMnd
96	67, 95	mpbid 222	. . . . . . 7 $/\$ $/\$ $\$ $/\$ ++ ++ Word Word freeMnd freeMnd
97		wrd0 13330	. . . . . . . . . . . 12 Word
98		ffvelrn 6357	. . . . . . . . . . . 12 Word Word $/\$ Word Word
99	66, 97, 98	sylancl 694	. . . . . . . . . . 11 $/\$ $/\$ $\$ $/\$ ++ ++ Word
100		lencl 13324	. . . . . . . . . . 11 Word
101	99, 100	syl 17	. . . . . . . . . 10 $/\$ $/\$ $\$ $/\$ ++ ++
102	101	nn0cnd 11353	. . . . . . . . 9 $/\$ $/\$ $\$ $/\$ ++ ++
103		0cnd 10033	. . . . . . . . 9 $/\$ $/\$ $\$ $/\$ ++ ++
104	102	addid1d 10236	. . . . . . . . . 10 $/\$ $/\$ $\$ $/\$ ++ ++
105	97, 13	syl5eleqr 2708	. . . . . . . . . . . 12 $/\$ $/\$ $\$ $/\$ ++ ++
106		oveq1 6657	. . . . . . . . . . . . . . 15 ++ ++
107	106	fveq2d 6195	. . . . . . . . . . . . . 14 ++ ++
108		fveq2 6191	. . . . . . . . . . . . . . 15
109	108	oveq1d 6665	. . . . . . . . . . . . . 14 ++ ++
110	107, 109	eqeq12d 2637	. . . . . . . . . . . . 13 ++ ++ ++ ++
111		oveq2 6658	. . . . . . . . . . . . . . . 16 ++ ++
112		ccatlid 13369	. . . . . . . . . . . . . . . . 17 Word ++
113	97, 112	ax-mp 5	. . . . . . . . . . . . . . . 16 ++
114	111, 113	syl6eq 2672	. . . . . . . . . . . . . . 15 ++
115	114	fveq2d 6195	. . . . . . . . . . . . . 14 ++
116		fveq2 6191	. . . . . . . . . . . . . . 15
117	116	oveq2d 6666	. . . . . . . . . . . . . 14 ++ ++
118	115, 117	eqeq12d 2637	. . . . . . . . . . . . 13 ++ ++ ++
119	110, 118	rspc2va 3323	. . . . . . . . . . . 12 $/\$ $/\$ ++ ++ ++
120	105, 105, 67, 119	syl21anc 1325	. . . . . . . . . . 11 $/\$ $/\$ $\$ $/\$ ++ ++ ++
121	120	fveq2d 6195	. . . . . . . . . 10 $/\$ $/\$ $\$ $/\$ ++ ++ ++
122		ccatlen 13360	. . . . . . . . . . 11 Word $/\$ Word ++
123	99, 99, 122	syl2anc 693	. . . . . . . . . 10 $/\$ $/\$ $\$ $/\$ ++ ++ ++
124	104, 121, 123	3eqtrrd 2661	. . . . . . . . 9 $/\$ $/\$ $\$ $/\$ ++ ++
125	102, 102, 103, 124	addcanad 10241	. . . . . . . 8 $/\$ $/\$ $\$ $/\$ ++ ++
126		fvex 6201	. . . . . . . . 9
127		hasheq0 13154	. . . . . . . . 9
128	126, 127	ax-mp 5	. . . . . . . 8
129	125, 128	sylib 208	. . . . . . 7 $/\$ $/\$ $\$ $/\$ ++ ++
130	59, 59	pm3.2i 471	. . . . . . . 8 freeMnd $/\$ freeMnd
131	50	frmd0 17397	. . . . . . . . 9 freeMnd
132	77, 77, 78, 78, 131, 131	ismhm 17337	. . . . . . . 8 freeMnd MndHom freeMnd freeMnd $/\$ freeMnd $/\$ Word Word $/\$ Word Word freeMnd freeMnd $/\$
133	130, 132	mpbiran 953	. . . . . . 7 freeMnd MndHom freeMnd Word Word $/\$ Word Word freeMnd freeMnd $/\$
134	66, 96, 129, 133	syl3anbrc 1246	. . . . . 6 $/\$ $/\$ $\$ $/\$ ++ ++ freeMnd MndHom freeMnd
135		eqid 2622	. . . . . . . . . 10 var_FMnd var_FMnd
136	135	vrmdf 17395	. . . . . . . . 9 var_FMnd Word
137	57, 136	ax-mp 5	. . . . . . . 8 var_FMnd Word
138		fcompt 6400	. . . . . . . 8 Word Word $/\$ var_FMnd Word var_FMnd var_FMnd
139	66, 137, 138	sylancl 694	. . . . . . 7 $/\$ $/\$ $\$ $/\$ ++ ++ var_FMnd var_FMnd
140	135	vrmdval 17394	. . . . . . . . . 10 $/\$ var_FMnd
141	61, 140	sylan 488	. . . . . . . . 9 $/\$ $/\$ $\$ $/\$ ++ ++ $/\$ var_FMnd
142	141	fveq2d 6195	. . . . . . . 8 $/\$ $/\$ $\$ $/\$ ++ ++ $/\$ var_FMnd
143	142	mpteq2dva 4744	. . . . . . 7 $/\$ $/\$ $\$ $/\$ ++ ++ var_FMnd
144	139, 143	eqtrd 2656	. . . . . 6 $/\$ $/\$ $\$ $/\$ ++ ++ var_FMnd
145	50, 77, 135	frmdup3lem 17403	. . . . . 6 freeMnd $/\$ $/\$ Word $/\$ freeMnd MndHom freeMnd $/\$ var_FMnd Word freeMnd _g
146	60, 61, 64, 134, 144, 145	syl32anc 1334	. . . . 5 $/\$ $/\$ $\$ $/\$ ++ ++ Word freeMnd _g
147	37, 52, 146	3eqtr4rd 2667	. . . 4 $/\$ $/\$ $\$ $/\$ ++ ++
148	4, 2, 1	mrsubff 31409	. . . . . . 7
149		ffn 6045	. . . . . . 7
150	148, 149	syl 17	. . . . . 6
151	150	adantr 481	. . . . 5 $/\$ $/\$ $\$ $/\$ ++ ++
152		fvex 6201	. . . . . . . 8 mREx
153	2, 152	eqeltri 2697	. . . . . . 7
154		elpm2r 7875	. . . . . . 7 $/\$ $/\$ $/\$
155	153, 56, 154	mpanl12 718	. . . . . 6 $/\$
156	48, 49, 155	sylancl 694	. . . . 5 $/\$ $/\$ $\$ $/\$ ++ ++
157		fnfvelrn 6356	. . . . 5 $/\$
158	151, 156, 157	syl2anc 693	. . . 4 $/\$ $/\$ $\$ $/\$ ++ ++
159	147, 158	eqeltrd 2701	. . 3 $/\$ $/\$ $\$ $/\$ ++ ++
160	159	ex 450	. 2 $/\$ $\$ $/\$ ++ ++
161	11, 160	impbid2 216	1 $/\$ $\$ $/\$ ++ ++

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 196 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wral 2912

cvv 3200 $\$ cdif 3571

cun 3572

wss 3574

c0 3915

cif 4086

cmpt 4729

crn 5115

ccom 5118

wfn 5883

wf 5884

cfv 5888 (class class class)co 6650

cmap 7857

cpm 7858

cc0 9936

caddc 9939

cn0 11292

chash 13117 Word cword 13291 ++ cconcat 13293

cs1 13294

cbs 15857

cplusg 15941

_g cgsu 16101

cmnd 17294 MndHom cmhm 17333 freeMndcfrmd 17384 var_FMndcvrmd 17385 mCNcmcn 31357 mVRcmvar 31358 mRExcmrex 31363 mRSubstcmrsub 31367

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-word 13299 df-lsw 13300 df-concat 13301 df-s1 13302 df-substr 13303 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-0g 16102 df-gsum 16103 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-frmd 17386 df-vrmd 17387 df-mrex 31383 df-mrsub 31387

This theorem is referenced by: mrsubco 31418

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